Step by step
The question
Question: The area of a circle grows at 20 cm²/s.
How fast is the radius growing when the radius is 5 cm?
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Draw and name
Draw it.
The radius r grows, so the area A grows too.
The area grows by 20 cm² per second: dA/dt = 20.
dr/dt means how fast r changes: the change each second.
We want dr/dt at the moment when r = 5.
2 Write the equation
The area of a circle: A = π r squared.
Careful!
Do not put r = 5 in yet!
r changes with time.
Numbers go in only after the derivative.
The right way: A = πr².
3 Take the derivative
Both A and r change with time.
The chain rule: r squared gives 2r times dr/dt.
π stays.
4 Put the numbers in
Put in dA/dt = 20 and r = 5.
Multiply the numbers: 2 × 5 = 10.
5 Solve for the rate
To remove times 10π, divide both sides by 10π.
Simplify: 20 and 10 both divide by 10.
That leaves 2 over π.
With π ≈ 3.14159, about 0.64 cm per second.
6 Check
Let's check: a tiny moment later, 0.001 seconds.
The area is 0.02 bigger.
Then r is about 5.000637.
r grew by 0.000637.
That is about 0.64 cm per second.
Right!
7 Pick the choice
Our answer is The radius grows at about 0.64 cm/s..
That is choice D.
8 Why not the others
A is 10/π.
Wrong: used r = 1: the radius is 5.
B is 4/π.
Wrong: lost the 2: dA/dt = 2πr × dr/dt.
C is 200π.
Wrong: multiplied: divide by 2πr.
Let's look back.
First: draw and name.
Then: write the equation.
Then: take the derivative.
Then: put the numbers in.
Then: solve for the rate.
Last: check.